The overall goal of this research program is the development and application of mathematical and statistical methods that can be applied to genetic data from human and non-human populations with the intention of understanding the past and present forces that determine their genetic composition. The kinds of data that will be considered are DNA sequences from different individuals or distinguishable genetic variants such as alleles at microsatellite loci, a relatively new and extremely useful class of genetic markers. This proposal will focus on four interrelated classes of problems: (l) the mutation process at microsatellite loci, (2) the history of individual alleles, including disease-causing alleles, (3) the extent of variation among disease-causing alleles at a single genetic locus, (4) the degree of non-random association among alleles at closely linked genetic loci. Microsatellite loci are of considerable practical interest because they are extremely abundant and highly variable in humans and other mammals. As a result, they are widely used in genetic mapping and in the analysis of population structure. Yet their use for these purposes depends on untested assumptions about the process of mutation. Further theoretical work is needed before population-level variation can be used to test hypotheses about the mutation process. Several parts of this proposal address the problem of the nonrandom association, or linkage disequilibrium, of closely linked genes, including genes that are closely linked to disease- causing loci. The extent of linkage disequilibrium depends in a complex way on various factors, including mutation, natural selection, migration among populations, and the history of population sizes. Mathematical models are needed that predict the extent of linkage disequilibrium under different conditions. Such models are needed now because of the widespread use of linkage disequilibrium for mapping disease-causing genes in human populations. The proposed research will be carried out using a combination of analytic theory, particularly the theory for Markov processes, and computer simulations. Many of the results will be in terms of mathematical likelihood functions that have desirable properties for estimating parameters and for testing hypotheses.